There are several links between climate change (of any origin) and the spread of pathogens. Not mutation rate that I know of, but the ability to break out of existing 'reserve' locations which persistently harbor them into areas where they're not normally found.
Case in point: the plague. Still very much alive in some areas of Asia and Africa, and localized outbreaks still occur today. But displacement of carriers species (e.g. rats) from those areas to others due to various climate-shift-induced pressures (such as too much or too little food, or human intrusion) can trigger outbreaks far and wide. The original plague period and spread was in many ways a perfect storm of factors but a re-occurrence is feasible (albeit today's improved knowledge of the cause, hygiene and containment measures should prevent it reaching anything like the scale once seen).
Another example: A few years back Myrtle Rust made its way over to NZ for the first recorded time, and one theory of how it got here was being carried on unusually warm/humid air currents from Australia which allowed it, as a kind of mold, to survive the journey.
Another vector: Melting permafrost releasing preserved pathogens to which today's populations (human and otherwise) have no immunity is another 'fun' climate change scenario. I find that one less plausible mostly because it would require human intrusion or some other means of transmission from the thawing site to hosts. But eh, maybe.